MinT - Best Known (38−26, 38, s)-Nets in Base 64

Best Known (38−26, 38, s)-Nets in Base 64

(38−26, 38, 177)-Net over F₆₄ — Constructive and digital

Digital (12, 38, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 38, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(38−26, 38, 257)-Net over F₆₄ — Digital

Digital (12, 38, 257)-net over F₆₄, using

net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
  - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(38−26, 38, 259)-Net in Base 64 — Constructive

(12, 38, 259)-net in base 64, using

2 times m-reduction [i] based on (12, 40, 259)-net in base 64, using
- base change [i] based on digital (2, 30, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(38−26, 38, 321)-Net in Base 64

(12, 38, 321)-net in base 64, using

2 times m-reduction [i] based on (12, 40, 321)-net in base 64, using
- base change [i] based on digital (2, 30, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(38−26, 38, 17120)-Net in Base 64 — Upper bound on s

There is no (12, 38, 17121)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 431 590184 952449 232491 472846 933552 646444 090424 702926 091495 321741 637664 > 64³⁸ [i]